The quantities v ,6 , F, and G in the forcing equations now have the 

 ^ o o 



respective values for the I" section: 



v" = V ; 6" = e" ; F" = - P" ; G" = - M" 

 o o 



Here F" and G" are the negatives of P" and M" , which act on the slice; and 



V is the transverse displacement of the beam at the slice. 



The H' section, however, is forced on the end (AA' ) facing positive 



X. A "beam whose forced end faces negative x, as in the previously 



developed theory, can be brought into the ^' position by a rotation 



through 180 degrees about an axis parallel to F. This rotation, however, 



reverses the spatial directions of 6^ and G. Hence, for the ^' section: 



v' = V ; e' = - e' ; F' = P' ; G' = - M' 

 o o 



(Note that G becomes -G by the rotation, but -G' = M' . ) 



r n . * 



Substitution into Equations LlTa,bJ then gives: 



F = F' + F" ; G = - G' + G" [l9a,b] 



These equations relate the actual external force and moment F and G to the 

 quantities which were denoted by F and G in the equations for forcing at 

 one end as applied to the two sections. 



Now, likewise, let the b's for the ?, ' and 2," sections be distinguished 



by one or two primes. Then Equations [l3a,b] of the general end- forcing 



theory become, for the two sections: 



F' = b' V + b' e ' ; F" = b'' v + b" e ^ 

 11 12 o 11 12 o 



G' = b' V + b' e' ; G" = b" V + b" 6" 

 21 22 o 21 22 o 



* Compare with Equation [19] of Reference 12 where G = 0. Note the use 

 of two coordinate systems (Figures T and 8) in that reference. 



23 



